shear heating induced thermal pressurization during earthquake nucleation · 2019. 11. 12. · r t_...

Shear heating‐induced thermal pressurizationduring earthquake nucleation

S. V. Schmitt,1 P. Segall,1 and T. Matsuzawa2

Received 3 October 2010; revised 2 March 2011; accepted 28 March 2011; published 23 June 2011.

[1] We model earthquake nucleation (in 2D) on narrow faults with coupled rate‐statefriction and shear heating‐induced thermal pressurization, including diffusive transport ofheat and pore pressure. Thermal pressurization increases pore pressure p, decreasingfrictional resistance. Observed fault core permeability is generally too low to mitigatethermal pressurization at subseismic slip speeds. Under drained, isothermal conditions,nucleation with the aging law is crack like, with the interior of the slip zone always nearmaximum slip speed. When thermal pressurization is included, it can dominate weakeningat speeds of 0.02–20 mm/s for hydraulic diffusivities chyd from 10

−8 to 10−3 m2/s andnominal material parameters well before seismic radiation occurs. Dramatic along‐strikelocalization of slip occurs due to feedback in which the area of maximum slip experiencesthe greatest weakening, which in turn favors more slip. With the slip law, however,nucleation is pulse like, with slip speed decaying behind the pulse tip. Thermalpressurization is diminished relative to the aging law case since most weakening occurs inlocations with limited slip, yet we find that it can overwhelm frictional weakening at slipspeeds in the range of 1–100 mm/s for chyd from 10

−8 to 3 × 10−5 m2/s. At higher slipspeeds, the finite thickness of the shear zone becomes significant, reducing thermalpressurization. Even if not the dominant weakening mechanism, thermal pressurization islikely to be significant at or before the onset of seismic radiation.

Citation: Schmitt, S. V., P. Segall, and T. Matsuzawa (2011), Shear heating‐induced thermal pressurization during earthquakenucleation, J. Geophys. Res., 116, B06308, doi:10.1029/2010JB008035.

1. Introduction

[2] Within the last 30 years, much attention has beendirected toward the role of rock friction in the nucleation ofearthquakes. According to the effective stress principle, thestrength of a fault is given by t = m(s − p), where m is thecoefficient of friction, s is the normal stress, and p is the porepressure. Laboratory and theoretical work has yielded rate‐and state‐dependent friction laws (starting with Dieterich[1979]), in which m depends on slip rate and slip history.Rate‐ and state‐dependent friction laws have provided plau-sible mechanisms for earthquake nucleation that are consis-tent with realistic seismicity rates [Ruina, 1983; Dieterich,1992, 1994]. At the same time, researchers [Sibson, 1973;Lachenbruch, 1980] realized that shear heating‐inducedthermal pressurization could lead to a loss of frictionalresistance during dynamic slip for constant m. The thermalpressurization hypothesis has subsequently been developedby many researchers [Mase and Smith, 1985, 1987; Lee andDelaney, 1987; Andrews, 2002; Bizzarri and Cocco, 2006a,2006b; Suzuki and Yamashita, 2006; Rice, 2006], but it has

been analyzed mainly in the context of fast slip. In this paper,we include thermal pressurization in numerical simulations ofearthquake nucleation.[3] The idea of shear heating‐induced thermal pressuri-

zation was first proposed in the context of faulting by Sibson[1973], though it had earlier been proposed as a mechanismfor catastrophic landslides [Habib, 1967]. Early work wasmotivated by a discrepancy between the theoretical amountof heat generated in earthquakes and the relative scarcity ofgeologic and geophysical evidence for such thermal sig-natures. The rate of frictional heat generation is proportionalto the product of stress and strain rate such that, neglectingconduction, DT / R t _� dt. With typical crustal normalstresses, fixed hydrostatic pore pressure, and narrow shearzones, the large strain rates in shear zones during earthquakerupture will generate sufficient heat to melt rock. Yetworldwide studies of exhumed fault zones seem not to showthe expected ubiquity of friction melt pseudotachylite [e.g.,Sibson, 1986]. Further, heat flow studies show no discern-ible thermal anomalies along active faults (for example,reporting on the San Andreas Fault [Lachenbruch and Sass,1980]). Sibson [1973] addressed this discrepancy by sug-gesting that thermal expansion of pore fluids weakens faultsduring earthquake rupture such that extreme heating doesnot occur.[4] In the following decades, researchers further examined

the thermal pressurization hypothesis. Starting with the

1Department of Geophysics, Stanford University, Stanford, California,USA.

2National Research Institute for Earth Science and Disaster Prevention,Tsukuba, Japan.

Copyright 2011 by the American Geophysical Union.0148‐0227/11/2010JB008035

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, B06308, doi:10.1029/2010JB008035, 2011

B06308 1 of 25

http://dx.doi.org/10.1029/2010JB008035

governing equations of heat, fluid flow, and dilatancy,Lachenbruch [1980] identified the conditions (includingshear stress, strain rate, slip speed, and permeability) thatallow heat and pore fluid transport to be neglected inmodeling fault zone strength during slip. He found thatrealistic values require solving coupled equations of friction,heat, and pore pressure transport, although he analyzed onlylimiting cases. Mase and Smith [1985, 1987] performednumerical simulations of shear heating using the coupledequations with an imposed slip history and a constant fric-tion coefficient. Andrews [2002] applied thermal pressuri-zation to a propagating crack model of an earthquake sourcewith slip‐weakening friction. Rice [2006] compared shearfracture energies inferred from seismological observations tothose predicted for a simple model of thermal pressurizationwith a constant friction coefficient and an imposed slip rate.Seismological estimates of fracture energy are influenced bymodel assumptions but independently yield values consis-tent with predictions of thermal pressurization, stronglysuggesting that thermal pressurization is relevant to naturalearthquakes. Noda et al. [2009] presented detailed numeri-cal calculations including rate‐state friction and thermalpressurization for artificially nucleated dynamic events, andshowed that model ruptures have many characteristics incommon with natural earthquakes.[5] Despite the concurrent developments in thermal

pressurization and rate‐ and state‐dependent friction, therehas been little effort to identify how they interact over anentire earthquake event from rupture nucleation to termi-nation. Conventional wisdom has been that thermal pres-surization occurs only in moderate to large earthquakes[Kanamori and Heaton, 2000; Andrews, 2002]. The originof this notion is unclear, but it may arise from Lachenbruch’s[1980] statement that thermal pressurization may occur onlyin large earthquakes provided that fault zone permeability isnear ∼10−15 m2. While rock permeability varies over manyorders of magnitude, recent laboratory studies of actual faultzone materials under appropriate confining pressures suggestthat fault core permeabilities are much lower than this value[Lockner et al., 2000; Wibberley and Shimamoto, 2003].[6] Indeed, recent work suggests that thermal pressuri-

zation is significant during earthquake nucleation. Sleep[1995a, 1995b] realized the possibility for significant shearheating with rate and state friction in certain conditions, butdid not consider the effects of fluid diffusion. Segall andRice [1995] considered shear dilatancy and the resultingpore pressure effect on earthquake nucleation, and the sameauthors later [Segall and Rice, 2006] considered thermalpressurization. In the latter work, they estimate that thermalpressurization may overwhelm rate/state frictional weaken-ing at slip speeds of 10−5 to 10−2 m/s, which are attainedwhen an earthquake’s moment release is still ten orders ofmagnitude smaller than those Andrews [2002] predicted(using a much larger value of hydraulic diffusivity, however).[7] Segall and Rice [2006] present a simplified analysis to

estimate when thermal pressurization is significant. Theystart with the slip history for an aging law nucleation zone offixed length [Dieterich, 1992] and estimate the temperaturerise that it would generate. Then, using hydraulic diffusiv-ities inferred from fault cores of the Median Tectonic Line[Wibberley and Shimamoto, 2003] and the Nojima Fault[Lockner et al., 2000], they predict the resulting pressuri-

zation rate. A critical velocity vcrit is defined to be whenthermal pressurization dominates rate/state friction in theoverall weakening. Assuming hydraulic diffusivity on theorder of 10−6 m2/s, Segall and Rice [2006] obtained a valueof vcrit ≈ 10−4 to 10−3 m/s, which is well below seismic slipspeeds and attained after only a modest amount of slip.Because of the coupled nature of the problem, there isfeedback in which slip leads to greater shear heating, which inturn leads to greater pore fluid pressurization that enhancesthe weakening and allows for even more slip. The analysisof Segall and Rice [2006] neglects this feedback and alsosimplifies the elastic interaction of an evolving nucleationzone by assuming a fixed nucleation zone size.[8] Our present work builds upon the analysis of Segall

and Rice [2006]. Using numerical models of predomi-nantly zero‐width faults embedded in a 2D diffusivedomain, we explore the behavior of the coupled system ofrate‐ and state‐dependent friction, shear heating‐inducedthermal pressurization, and pore pressure diffusion. Bysolving the system numerically in an elastic diffusive con-tinuum, we include the thermal weakening feedback processand allow the nucleation zone to evolve in size. The maingoal of this work is to identify when thermal pressurizationdominates fault weakening, which will ultimately influencethe initial conditions of dynamic ruptures.

2. Model and Mathematical Description

2.1. Model Geometry and Physical Basis

[9] The fault model is motivated by recent observations offault zones in which slip appears to be hosted within anarrow, low‐permeability fault core (Figure 1a). That con-ceptual model is based on studies of two Japanese faults andtwo Southern California faults. Wibberley and Shimamoto[2003] measured the permeability of fault core materialfrom the Median Tectonic Line (MTL) in Japan, and found alow permeability (∼10−19 m2 at 80–180 MPa confiningpressure) fault core of width ∼0.1 m surrounded by a higher‐permeability (∼10−16 to ∼10−14 m2) damage zone. Lockneret al. [2000] measured the permeability of drill core sam-ples from 1500 m depth on the Nojima fault. They foundsimilar properties at 50 MPa confining pressure: a narrowfault core with permeability of ∼10−19 to ∼10−18 m2 and asurrounding damage zone with permeability of ∼10−17 to∼10−16 m2.[10] The width of the actively shearing zone is a key

parameter in the analysis of thermal pressurization. Geolo-gists have undertaken several field studies to quantify thethickness of shear zones active during seismic slip (summa-rized by Sibson [2003]). Shear zone thicknesses of 1–10 mmhave been observed in slickenside material, pseudotachy-lites, and intravein septa in laminated quartz veins (thinlayers of wall rock thought to be remnants of slip on the veinmargin). Gouge shear zones appear to be wider, withthicknesses of 3–300 mm. The distribution of shear straininside shear zones is not as well documented. Thin sectionanalysis of samples from the North Branch San Gabriel fault[Chester et al., 1993] and the Punchbowl fault [Chesterand Chester, 1998] suggest that extreme shear localizationmay occur during earthquakes. These authors document a0.6–1.1 mm apparent shear zone, within which most slipoccurs in a zone of width 100–300 mm [Rice, 2006]. No

SCHMITT ET AL.: THERMAL PRESSURIZATION DURING EARTHQUAKE NUCLEATION B06308B06308

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signatures of extreme shear localization were documented inthe MTL sample, but Rice [2006] reports that the researchersobserved a

which relates shear strength tfric to effective normal stressseff = (s − p) through a friction coefficient m. The frictioncoefficient m depends on slip rate v and state �, which hasdimension of time and evolves over characteristic distance dc.The parameters a and b are dimensionless material constantson the order of 0.01. We analyze the two forms of stateevolution law in common use: one referred to as the “aginglaw,” and the other referred to as the “slip law.” They are[Ruina, 1983]

d�

dt¼ 1� �v

dc; the aging law; and ð5Þ

d�

dt¼ � �v

dcln�v

dc; the slip law: ð6Þ

For both laws, the steady state value is �ss = dc/v, but the twostate evolution laws yield different behavior in response toan abrupt change in sliding velocity. Following a stepincrease in velocity, the aging law predicts evolution offriction toward steady state over a slip distance proportionalto the velocity change, while slip law friction predicts acharacteristic evolution distance that is independent of thevelocity change. Ampuero and Rubin [2008] demonstratedthat, under drained, isothermal conditions (fixed seff), thetwo state evolution laws yield profoundly different nucle-ation behaviors. They found that aging law nucleation takesthe form of a quasi‐statically growing crack for 0.5 ⪅ a/b < 1or a fixed length slip zone for a/b < 0.5, while slip lawnucleation is more pulse like. Laboratory velocity‐steppingexperiments tend to exhibit behavior more consistent withthe slip law [Nakatani, 2001]; for that reason Ampuero andRubin [2008] favor the slip law in the context of frictionalnucleation. Laboratory experiments with surfaces in sta-tionary contact, however, favor the aging law [Dieterich andKilgore, 1994].[15] Linker and Dieterich [1992] found that variable

effective normal stress seff(t) also affects the coefficient offriction m. This effect is relevant to the present work becauseeffective stress varies due to the thermal pressurization.Linker and Dieterich [1992] defined a constitutive rela-tionship in which the variable normal stress affects thefrictional state,

d�

dt¼ state evolution

law 5ð Þor 6ð Þ� �

� ��b�eff

d�effdt

; ð7Þ

where a is a coefficient in the range of 0 to m0. For a = 0, thenormal stress effect vanishes; that is, a step change in seffresults in an instantaneous change of t to its new value.Increasing a decreases the instantaneous response. With a =m0, a step change in seff results in a gradual evolution towardthe final value of t with no instantaneous change. Linker andDieterich [1992] and Richardson and Marone [1999] findvalues of a ≈ 0.3 agree with laboratory experiments.2.2.2. Thermal Transport[16] For spatially uniform properties, the equation for

thermal diffusion with a shear heating source (neglectingdiffusion parallel to the fault) is

�d�

dtþ kth @

2T

@y2¼ �cv @T

@t; ð8Þ

where g is shear strain, r is the bulk density of the rock andpore fluid, kth is thermal conductivity, and cv is heat capacity(a typical value of rcv is 2.8 MPa/K). Outside of the shearzone we neglect shear heating, so equation (8) reduces tohomogeneous diffusion equation

@T

@t¼ cth @

2T

@y2for y > 0 ; ð9Þ

where cth = kth/rcv is the thermal diffusivity. Typical valuesof cth are very near 10

−6 m2/s [Rice, 2006]. For heat gen-eration and transport within the shear zone, we considerenergy conservation over a finite width shear zone extend-ing from y = −h to y = 0:

� hd�

dt

� �þ 2kth @T

@y

��y¼0

� �cvh @T@t

¼ 0 for �h

Reasonable values of L fall in the range 0.6–1.1 MPa/K[Segall and Rice, 2006]. Inside the shear zone, fluid massconservation requires

@p

@t¼ L @T

@t� 2chyd

h

@p

@y

��y¼0

: ð15Þ

In the case of the zero‐width shear zone, ∂p/∂y∣y = 0 = 0. For ahomogeneous medium with diffusion perpendicular to azero‐width fault, Rice [2006, Appendix B4] found that porepressure on the fault (that is, at y = 0) is directly related to thetemperature rise through the relationship

Dp y¼0ð Þ ¼ L1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffichyd=cthp DT y¼0ð Þ : ð16Þ

We leverage this relationship to yield a significant reductionin computational expense, since in the limit h→ 0 it allows usto simulate numerically the temperature field while directlycomputing Dp(y = 0), rather than simultaneously computingthermal and pore pressure diffusion.2.2.4. Stress Interaction, Seismic Radiation, andGoverning Equation for Slip[18] In modeling earthquake nucleation in a continuum,

we must calculate the elastic stress that arises from slip onthe fault. For a 1D fault in a 2D continuum, the stressinteraction is computed as a Hilbert transform of the dis-placement gradient. For antiplane deformation,

�el xð Þ ¼ �∞ � G2Z ∞�∞

1

x� �@�

@�d� ; ð17Þ

where t∞ is the remote (tectonic) stress. For plane straindeformation in equation (17), G is replaced by G/(1 − n),where n is Poisson ratio. The Fourier transform of (17) canbe written as

�̂el mð Þ ¼ �̂∞ � k̂ mð Þ �̂ ; ð18Þ

where k̂(m) is an effective stiffness that depends on spatialwave number m (for antiplane deformation, k = G∣m∣/2)[e.g., Segall, 2010]. Determining the elastic stress interac-tion in the Fourier domain using an FFT is computationallyefficient.[19] As slip speeds grow to within a few orders of mag-

nitude of elastic wave speeds, inertial effects becomeimportant. Because our focus is on nucleation, we employthe “radiation damping” approximation [Rice, 1993] ratherthan full elastodynamics [e.g., Lapusta et al., 2000]. Radi-ation damping is given by

�rad ¼ G2vs v ; ð19Þ

which gives the stress change due to radiation of plane swaves.[20] With equations for elasticity (17), friction (4), and

radiation (19), we write the governing equation for the faultslip as tel − tfric − trad = 0, which expands to

�∞ � G2Z ∞�∞

1

x� �@�

@�d� � �0 þ a ln vv0 þ b ln

�v0dc

� �� p tð Þ½ �

� G2vs

v ¼ 0 : ð20Þ

The effects of thermal pressurization act through the porepressure term in tfric. The weakening rates due to frictionand thermal pressurization become separate terms in thetime derivative of the force balance (20), which is

_�∞ � G2Z ∞�∞

1

x� �@v

@�d� � �� p tð Þ½ � a

v

dv

dtþ b�

d�

dt

� �

þ �0 þ a ln vv0 þ b ln�v0dc

� �dp

dt� G2vs

dv

dt¼ 0 : ð21Þ

In this study, we assume parameters consistent with a depthof 7 km; their nominal values are in Table 1. Hydraulicproperties are motivated by the observations from Japan[Lockner et al., 2000; Wibberley and Shimamoto, 2003].The frictional parameter (b − a) is consistent with the ob-servations of Blanpied et al. [1995], while the value of dc isset at the high end of laboratory observations at 100 mm.This value of dc does favor thermal pressurization relative todc ∼ 10 mm, though we explore the dependence on dc insections 2.3 and 5.4.[21] To estimate when radiation damping becomes

important, we estimate the velocity at which it balances thevelocity dependence of rate/state friction in isothermalnucleation. Momentarily neglecting thermal pressurization,we rearrange equation (21) to obtain

a

vþ G2�eff vs

� �dv

dt¼ 1�eff

d�eldt

� b�

d�

dt: ð22Þ

Table 1. Notation

Symbol Quantity Nominal Value or Reference

m0 nominal value of friction 0.6a velocity dependence of friction 0.012b state dependence of friction 0.015a dependence of � on _�eff 0 to 0.6dc characteristic frictional slip

weakening distance10−4 m

G shear modulus 10 GPavs shear wave speed, for radiation

damping3700 m/s

cth thermal diffusivity 10−6 m2/s

chyd pore pressure diffusivity 10−6 m2/s

L thermal pressurization factor 0.8 MPa/Krcv volumetric heat capacity 2.86 MPa/Ks − p0 background effective stress 140 MPa_�∞ stressing rate 0.01 Pa/sv0 friction reference velocity Equation (4)Lmin minimum half length for unstable

nucleation zoneEquation (25)

L∞ maximum aging law nucleation zonehalf length

Equation (35)

h shear zone thickness Sections 2.1 and 6.3m friction coefficient Equation (4)v slip velocity Equation (20)� frictional state Equation (5) and (6)vmax(t) maximum sliding velocity at time tvcrit vmax when ∣m0 _p∣ > ∣ _�(s − p0)∣ Equation (37)T temperature Equations (9) and (11)p pore pressure Equations (12) and (16)seff effective normal stress s − p0 − ptnst time of the earthquake Sections 3.1, 4.1, and 4.2


5 of 25

From this it is apparent that radiation damping dominates atspeeds v > 2avsseff/G, which is about 0.9 m/s for thenominal values shown in Table 1. To identify conditions forwhich the quasi‐dynamic governing equation is valid, Riceet al. [2001] show that for slip speeds

v � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia b� að Þ

p �eff vsG

; ð23Þ

solutions are effectively equivalent to those of the elastody-namic equations. For our nominal values of system para-meters, quasi‐dynamic analysis is sufficient for v� 0.5 m/s.The speed at which radiation damping and elastodynamiceffects become important is proportional to the effectivenormal stress. The effective normal stress is not well known atseismogenic depths. For hydrostatic pore pressures, neglect-ing inertial terms is reasonable for v ⪅ 0.1 m/s, but both theradiation damping term and wave‐mediated effects wouldbe significant at much lower slip speeds for near‐lithostaticpore pressure.

2.3. Dimensional Analysis

[22] Writing the governing equations in a nondimensionalform gives insight into the effects of the various system para-meters. We define a nondimensional frictional strength as

�̂ ¼ �f ric�0 �� p0ð Þ ;

where p0 is the background pore pressure. In our simulations,we apply a constant remote stressing rate _�∞, which can berelated to a velocity through the nucleation zone stiffness k. Inthe spring‐slider analogue, linear stability analysis [Ruina,1983] reveals that small perturbations become unstable if thespring’s stiffness is less than a critical value given by

kcrit ¼ �eff b� að Þdc : ð24Þ

In an elastic continuum the stiffness is directly proportional tothe elastic modulus and inversely proportional to the halflength of the sliding zone, or k ≈ G/2L. Hence we can define acharacteristic length

Lmin ¼ Gdc2 b� að Þ �� p0ð Þ ; ð25Þ

which is the minimum half length of a nucleation zone capableof accelerating to instability [e.g., Rubin and Ampuero, 2005].We then obtain the nondimensional along‐fault coordinate

x̂ ¼ xLmin

¼ 2 b� að Þ �� p0ð ÞxGdc

: ð26Þ

A characteristic velocity can be defined based on the imposedstressing rate _�∞:

v∞ ¼ _�∞kcrit ¼dc

b� að Þ �� p0ð Þ _�∞ : ð27Þ

The velocity v∞ represents the slip speed in a hypotheticallocked zone with stiffness kcrit that experiences stress rate_�∞. Slip speed, time, and frictional state thus have sensiblenormalizations given by

v̂ ¼ vv∞

; t̂ ¼ tv∞dc

; and �̂ ¼ �v∞dc

:

For a homogeneous medium there is no characteristic dis-tance for heat and pore fluid transport away from the fault,so we define one to be the pore pressure diffusion lengthcorresponding to time dc/v∞. Thus, nondimensional distanceperpendicular to the fault is

ŷ ¼ yffiffiffiffiffiffiffiffiffiffiffiffiv∞

chyddc

r:

We define nondimensional temperature and pore pressurechanges by

T̂ ¼ TL�� p0 and p̂ ¼

p

�� p0 :

The nondimensional rate‐ and state‐dependent frictionalresistance (4) becomes

�̂ ¼ 1þ a�0

log v̂þ b�0

log �̂

� �1� p̂ð Þ ; ð28Þ

with state evolution laws (from 5–7)

d�̂

dt̂¼

1� �̂v̂þ ��̂b 1� p̂ð Þ

dp̂

dt̂; the aging law; or

��̂v̂ log �̂v̂

�

þ ��̂b 1� p̂ð Þ

dp̂

dt̂; the slip law:

8>>><>>>:

ð29Þ

The pore pressure diffusion equation (12) simplifies to

@p̂

@ t̂¼ @

2p̂

@ŷ2þ @T̂@ t̂

; ð30Þ

while the thermal diffusion equation (9) transforms to

@T̂

@ t̂¼ D @

2T̂

@ŷ2; ð31Þ

with D = cth/chyd. The Neumann boundary condition (11)becomes

@T̂

@ŷ

��ŷ¼0

¼ � �0L2�cv

ffiffiffiffiffiffiffiffiffiv∞dccth

r ffiffiffiffiffiffiffiffichydcth

r�̂ v̂ ¼ �FD�1=2�̂ v̂ ; ð32Þ

where the nondimensional group FD−1/2 quantifies theintensity of thermal pressurization. This group is equivalentto the “thermal pressurization efficiency” ET of Segall et al.[2010]. In terms of stressing rate,

F ¼ �0L2�cv

ffiffiffiffiffiffiffiffiffiv∞dccth

r¼ �0dcL

2�cv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_�

cth b� að Þ �� p0ð Þ

s:


6 of 25

The zero‐width shear zone p‐T relationship of equation (16)simplifies to

p̂ ŷ¼0ð Þ ¼ T̂ ŷ¼0ð Þ1þD�1=2 : ð33Þ

We thus identify five nondimensional groups in equations(28)–(33): a/m0, a/b, a/m0, D and F . The first two nondi-mensional groups are the well documented [e.g., Ampueroand Rubin, 2008] parameters governing rate‐ and state‐dependent frictional nucleation. The nondimensional groupa/m0 is from the Linker and Dieterich [1992] constitutivelaw; in this case it controls how quickly friction responds toa change in pore pressure. In some studies (for example, slipon a bimaterial fault [Cochard and Rice, 2000]) this effecthas important physical implications. In the present study,the effect is crucial for numerical well posedness (seeSection 3.4), but does little to affect the overall behavior ofthe system.[23] The diffusivity ratio D measures relative rates of heat

and pore fluid diffusion, and is the least well constrained ofthe parameters, due to uncertainty in permeability. Boththermal and fluid diffusion diminish shear heating‐inducedthermal pressurization. With chyd � cth, thermal conductiondominates diffusion away from the fault, while for chyd �cut pore fluid diffusion is the primary diffusive mechanism.Since chyd is the least well known of all of the materialproperties relevant to thermal pressurization, we study itseffect in detail (section 5.5).[24] The remaining nondimensional group F is a measure

of shear heating source strength: larger values of rcv (vol-umetric heat capacity) or cth diminish the temperature rise,while stronger system loading via v∞ or longer slip weak-ening distances dc will enhance shear heating. The thermalpressurization factor L enters F because of the scaling oftemperature on the fault with pore pressure.[25] There are two additional nondimensional groups

not analyzed in detail here. Since this study focuses on thezero shear zone width approximation, the ratio h/dc does notappear. Some laboratory work has been done to estimate h/dc;for example,Marone and Kilgore [1993] propose h = 100dc.A limited number of finite shear zone width calculationsare described in section 6. Another nondimensional group isW/Lmin, where W is the along‐strike dimension of the fault.For slip events that become large enough that the faultlength is important, or for which the nucleation dimension isa significant fraction of W, the ratio W/Lmin can be critical incontrolling the behavior (for example, the slow slip eventsof Liu and Rice [2005] or Segall [2010]). In the presentwork the fault length is large compared to the nucleationdimension, W � Lmin, which we expect to be the norm forearthquakes.

2.4. Numerical Implementation

[26] Using the zero‐width shear zone approximation, wecouple a 2D finite difference code with a boundary integralelasticity formulation to simulate nucleation with thermalpressurization. With the time derivative of the stress balance,equation (21), we can pose the problem in the form of asystem of coupled ODEs. We integrate v, �, and T using anexplicit Runge‐Kutta method (see Appendix A). For the short

integration times considered here, this approach introducesnegligible error.[27] The fault is modeled as one edge of a finite difference

grid used to calculate thermal diffusion. We impose theNeumann boundary condition (11) at the fault, whichintroduces the appropriate shear heating at y = 0. With azero‐width shear zone and homogeneous material properties,pore pressure may be obtained directly from the computedtemperature using equation (16). We calculate the elasticstress interaction in the Fourier domain using equation (18).[28] The finite difference grid adaptively remeshes as

accuracy demands. Early in nucleation, temperature gra-dients are slight, yet over long time intervals the temperatureperturbation extends far from the fault. As slip accelerates,however, the heat generation progressively outweighs ther-mal diffusion. Hence the largest temperature gradients occurnear the fault, and solution accuracy is limited by our abilityto correctly discretize those steep gradients. Therefore, whenaccuracy requires smaller grid point spacing away from thefault, we halve the grid spacing in the y dimension. In orderto yield acceptable numerical accuracy, our simulations takeexplicit time steps limited by the Courant‐Friedrichs‐Lewy(CFL) condition, Dt ≤ Dy2/2cth, though the ODE solvermethod’s automatic time step selections are usually moreconservative.[29] The system is driven by a constant remote stressing

_�∞ = 0.01 Pa/s (0.3 MPa/yr). We initiate simulations with allpoints on the fault sliding at tectonic velocities randomlydistributed near 10−9 m/s and below steady state (that is,v�/dc < 1). These initial conditions cause deceleration ofslip until sufficient stress accumulates to drive the faultback toward steady state. Thermal pressurization duringthis phase is negligible; we therefore avoid the computa-tional expense of diffusion by integrating only isothermalrate‐ and state‐dependent friction until the initial formationof a nucleation zone. Tests have shown that we may usethis simplification without any detectable influence on themuch later phase of the simulations when thermal pres-surization becomes significant.

3. Results: Aging Law Nucleation

3.1. Comparison to Isothermal, Drained Nucleation

[30] One of our main goals of this work is to examine hownucleation is affected by thermal pressurization. Beforeperforming such a comparison, we summarize some char-acteristics of isothermal, drained nucleation. For both formsof the state evolution law, the initial stage of nucleation islocalization of slip in a zone of half length Lmin, starting atspeeds that will be on the order of v∞. We shall call this the“initial weakening phase.” After some amount of slip occursin the initial weakening phase, nucleation behavior transi-tions to a different mode. Rubin and Ampuero [2005] foundthat for a/b ⪅ 0.5, acceleration proceeded on a fixed lengthzone of dimension

L� � 1:38 Gdcb�eff : ð34Þ

For 0.5 ] a/b < 1, Rubin and Ampuero [2005] found thatnucleation takes the form of a growing crack. That is, thenucleation zone grows wider as it accelerates, and the


7 of 25

interior of the crack continues to slip. Using fracturemechanics arguments, they show that the half widthasymptotically approaches

L∞ ¼ Gdc b

b� að Þ2�eff

: ð35Þ

We assume typical laboratory‐inferred values of a and bwitha/b = 0.8, which correspond to the crack‐like nucleationregime. Isothermal, drained simulations (shown Figure 2,left) agree with the results of Rubin and Ampuero [2005].After slip of ∼10dc (an amount that depends on _�∞ and the

initial conditions), the nucleation zone transitions from theinitial slip‐weakening phase to a growing crack approachingwidth 2L∞. Inside the nucleation zone, slip is nearly at steadystate, which can be seen in plots of log(v�/dc) in Figure 2b.As the nucleation zone accelerates, steady state velocityweakening scales with ln v, which is shown by the regularlyspaced snapshots of stress drop inside the nucleation zone inFigure 2c. This is understood by noting that at steady state, ln(v�/dc) = 0 and m(v, t) = m0 + (a − b) ln[v(t)/v0].[31] With shear heating‐induced thermal pressurization

included, aging law nucleation is markedly different (rightcolumn of Figure 2). In these simulations, we use typical

Figure 2. Comparison of aging law nucleation (a‐c) without and (d‐f) with thermal pressurization. Snap-shots of parameters on the fault are shown. The time interval between snapshots is not uniform; rather,snapshots are taken at every decade in slip speed in the middle of the nucleation zone. Figures 2a and2d show snapshots of slip speed on the fault. The qualitative difference above 10−4 m/s suggests whenthermal pressurization begins to dominate fault weakening inside the nucleation zone. Bars of width 2Lmin(bottom) and 2L∞ (top) are shown. Figures 2b and 2e are snapshots of log10(v�/dc), which is a measureof how far the fault is from frictional steady state. In isothermal, drained nucleation (Figure 2b), theinterior of the nucleation zone is slightly above steady state in these simulations. With thermal pressur-ization (Figure 2e), the interior of the nucleation zone is driven above steady state when thermal pressuri-zation dominates. Figures 2c and 2f are snapshots of frictional resistance to slip. In isothermal, drained aginglaw nucleation (Figure 2c), the stress drop inside the nucleation zone is nearly uniform. With thermal pres-surization (Figure 2f), the stress drop is not uniform inside the nucleation zone. Figure 2g shows slip speed inthe middle of the nucleation zone over time with and without thermal pressurization. Thermal pressurizationonly slightly advances the time of the instability. Dots correlate to the snapshots shown in Figures 2a–2f.


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values of the friction and diffusion parameters (though dc issomewhat large at 100 mm), but we exclude the effect ofvariable normal stress on frictional state (that is, a = 0). Atlow slip speeds (compare the snapshots of slip speed in plots[a] and [d]), the two cases are identical. As the maximumslip speed increases, however, nucleation becomes qualita-tively different. In those snapshots, slip speed ceases to benearly uniform in the middle of the nucleation zone; rather,it localizes along strike. The qualitative deviation fromisothermal, drained nucleation indicates a critical velocityvcrit above which thermal pressurization dominates theweakening of the fault. At even higher slip speeds, thelocalization of slip is dramatic. If allowed to progresswithout radiation damping, we find that the nucleation zonelength eventually tends toward zero, regardless of thenumerical discretization along the fault. This singularity innucleation zone dimension indicates that the physicalproblem is poorly posed, which we consider further insection 3.3.[32] Figures 2b and 2e compare the frictional state in the

two cases. As mentioned above, the interior of the nucle-ation zone in the isothermal, drained case is very near steadystate. With thermal pressurization, the interior of thenucleation zone is driven above steady state when v > vcrit.There, direct velocity strengthening (the a term) in rate/statefriction determines the frictional response.[33] Figures 2c and 2f compare the frictional resistance to

slip tfric in the two cases. With thermal pressurization, thestress drop inside the nucleation zone is not uniform. Thefeedback inherent in thermal pressurization is apparent inthe localization of the stress drop corresponding to thelocalization of slip. While difficult to see in Figure 2f, there

is also a small amount of healing (an increase in tfric) thatoccurs on the flanks of the localized slipping zone.[34] Thermal pressurization weakens the fault more than

rate‐and‐state friction alone, so we expect that it willadvance the time of the frictional instability tinst. Our simu-lations confirm that prediction, as shown in Figure 2g. Thetime advance is modest; in the simulations of Figure 2, theadvance of the instability is 8 × 102 s ahead of the 8 × 108 sbetween the start of the simulation and the frictional insta-bility in the isothermal case.[35] Since maximum slip and slip speed are colocated at

the middle of the nucleation zone in aging law nucleation(with or without thermal pressurization), it is instructive toexamine the contributions to the stress rate (equation (21))there in order to understand how thermal pressurizationaffects the frictional resistance. In Figure 3, we track theirevolution with slip. For slip of less than ∼20dc (2 mm),rate/state friction is the only significant weakening mech-anism, and it is entirely balanced by elasticity. As slipproceeds over the next ∼14dc (1.4 mm), rate/state frictioncontinues to dominate the fault weakening, but thermalpressurization increases in magnitude until it eventuallybecomes dominant. The crossover when ∣m0 _p∣ > ∣seff _�∣defines vcrit, which we examine in more detail in section 3.2.With thermal pressurization dominant, _� changes sign tobecome strengthening with about 3dc (0.3 mm) additionalslip. After that time, thermal pressurization drives the faultweakening such that _�/� is much less than the acceleration_v/v. Thus, the direct velocity effect (the a term in thefriction law [4]) causes friction _� to become strengthening;in other words, frictional state cannot “keep up” with theacceleration. We also note that radiation damping starts togrow in importance at a slip of ∼33dc, but that it is stillseveral orders of magnitude weaker than any other effectuntil ∼40dc, which corresponds to vmax ≈ 0.1 m/s.[36] During the nucleation phase, the fault accumulates

relatively modest slip and temperature rise by the time thatthermal pressurization dominates frictional weakening.Figure 4a illustrates profiles of cumulative slip corre-sponding to the same snapshots in Figure 2. The slip profilesare similar to isothermal, drained nucleation (not shown)with the exception of a small amount of localized slip visibleat the center of the nucleation zone in snapshots corre-sponding to high slip speeds. As inferred from Figure 3,thermal pressurization dominates the fault weakening afteronly 3.4 mm of slip has occurred. The corresponding tem-perature rise is only about 3.5°C (Figure 4b). With the p‐Trelationship for the zero‐width shear zone (16), that corre-sponds to a pore pressure increase of 1.4 MPa, which is only1% of the initial effective normal stress.

3.2. Critical Velocity

[37] We quantify the critical velocity for thermal pres-surization vcrit as the peak velocity on the fault at the timewhen ∣m0 _p∣ = ∣ _�(s − p0)∣. In defining vcrit, we neglect thesecond‐order quantities (m − m0) _p and _�Dp for ease ofanalysis and because they are small compared to m0 _p and_�(s − p0). For a single point, the definition of vcrit isstraightforward, but for a continuum there is some spatialvariability of which effect dominates the fault strength. We

Figure 3. Terms in the stress rate equation (21) at thecenter of the nucleation zone, plotted against cumulative slipat that location. Negative values indicate weakening. Thevertical scale is logarithmic in both directions away from thehorizontal axis (in the shaded area, values between –e and eare on a linear scale).


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therefore integrate the aforementioned terms over all regionson the fault that have weakened by either mechanism. Ateach time step, we compare the magnitudes of

Mtp ¼ �RL tð Þ �0

@p@t dx and

Mf ric ¼RL tð Þ �� p0ð Þ @�@t dx ;

ð36Þ

where L(t) is the region of x for which _� f is negative or wasnegative at times prior to t. That domain of integration issomewhat arbitrary, but a more physically motivated defi-nition of vcrit or weighting of the integrand would onlychange its value by a small factor. We define the criticalvelocity to be

vcrit � vmax when Mtp < Mf ric for the first time�

: ð37Þ

Figure 5 shows graphically the determination of vcrit. Withparameters used in Figure 2, the critical velocity is 6 ×10−5 m/s, which is far below seismic slip speeds.We note thatbefore vcrit is attained, thermal pressurization already dom-inates at the center of the nucleation zone. There, _�(s − p0) =−m0 _p at vmax = 3 × 10−5 m/s.

3.3. Shrinking Nucleation Zone

[38] As discussed in section 3.1, thermal pressurizationcauses the nucleation zone to shrink for slip speeds abovevcrit, and it eventually evolves to a singularity in nucleationlength. This can be understood by noting that _p / vmax3/2 . Thisrelationship was predicted by Segall and Rice [2006],equation 55 based on the rate of heat production ignoringthe change in frictional resistance due to pressurization, andis seen in Figure 5 here (see also section 5.2). Thus, _� / −v3/2,which implies that

@�

@v/ � v

3=2

_v: ð38Þ

In aging law nucleation, both v and _v are always positive, sofriction on the fault is directly velocity weakening oncethermal pressurization dominates. A linear stability analysis

near steady state [e.g., Segall 2010, pp. 335–336] shows thatfor pure velocity weakening friction a perturbation dv growsaccording to

�v tð Þ ¼ exp � kt@�@v

��v0

!; ð39Þ

where k is the stiffness of the loading system. In an elasticmedium, stiffness is inversely proportional to the length of theslipping zone and is always positive. Since the argument ofequation (39) is always positive, any velocity perturbationwill grow to instability, regardless of how large k (or small L)is. In short, elasticity alone cannot stabilize a shrinkingnucleation zone on a zero‐width fault weakening by thermalpressurization.

Figure 4. (a) Snapshots of cumulative fault slip during nucleation with thermal pressurization. Snap-shots correspond to those in Figure 2. (b) Snapshots of pore pressure and temperature on the fault,which are uniquely related through equation (16).

Figure 5. Evolution of the weakening rate within thenucleation zone with aging law friction and thermal pressur-ization. Critical velocity vcrit is shown by the vertical dashedline and is defined whenMtp = Mfric (see equation (36)). Thedotted line has a slope of −3/2 for reference.


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3.4. The Effect of Variable Normal Stress on FrictionalState

[39] In the simulations presented above, a change ineffective normal stress leads to an immediate change inshear stress on the fault. Linker and Dieterich [1992],however, observed in rock sliding experiments that changesin shear stress lag behind changes in normal stress. Forsliding at constant velocity with an imposed step change innormal stress, part of the response in shear stress is instan-taneous, with the remainder occurring over a small slipdistance: a few dc in the rate‐and‐state friction framework.Linker and Dieterich [1992] presented a constitutive law (7)that is also applicable to continuous changes in effectivenormal stress and is relevant to the present work.[40] We tested the influence of the Linker‐Dieterich effect

by conducting simulations with a = 0.3 and a = m0 = 0.6,keeping all other parameters fixed. Linker and Dieterich’s[1992] laboratory studies found values of 0.2 < a < 0.3,and the end‐member case of a = m0 serves to place an upper

bound on the influence of this effect. Figure 6 compares slipspeed and temperature profiles for the two values of a. Themost striking effect is that increasing a reduces the extremelocalization of slip. In both of the simulations shown inFigure 6, the nucleation zones did not collapse to a zero‐length singularity before the simulations’ ends. However,we have found that, given sufficient time, a < m0 alwaysleads to a zero‐length slip singularity provided that radiationdamping does not dominate first.[41] With a > 0, the temperature increase is amplified

(Figure 6). This is a direct consequence of the influence ofshear stress on heat production. Without the Linker‐Dietericheffect (a = 0), an incremental decrease in effective stress leadsimmediately to a decrease in shear strength. With a > 0,there is a delay in the loss of shear resistance. Hence, forsimilar values of v, the resulting shear stress on the fault isgreater, and more shear heating therefore occurs.[42] Despite the significant effect on nucleation zone

shape and fault zone temperature, we find that the Linker‐Dieterich effect does not significantly affect vcrit, the speed

Figure 6. Results of two simulations of aging law nucleation with thermal pressurization including theLinker and Dieterich [1992] effect. These simulations were started with identical initial conditions asthose of Figure 2, which neglects the effect (a = 0). (a) Snapshots of slip speed with a = 0.3. The activeslipping region shrinks less than with a = 0. (b) Snapshots of temperature and pressure the same times asFigure 6a. (c) Slip speed profiles with a = m0 = 0.6, showing even less localization. (d) Snapshots oftemperature and pressure corresponding with Figure 6c.


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at which thermal pressurization dominates. Yet its effect onthe nucleation zone geometry indicates that it (or a similareffect such as the one reported by Prakash and Clifton[1993]) must be included in fault models with an infinites-imally thin shear zone. A normal stress effect may not be ascritical, however, for models that incorporate a finite widthshear zone; the diminished temperature rise of such modelswill reduce the nucleation zone’s tendency toward extremelocalization.

4. Slip Law Nucleation

4.1. Comparison to Isothermal, Drained Nucleation

[43] Isothermal, drained nucleation with the slip law ismarkedly different than with the aging law. Near steadystate, both exhibit the same response to perturbations in slipspeed, so both have an initial slip‐weakening phase withlength scale Lmin. As slip accelerates, however, slip law

nucleation takes the form of a unidirectional slip pulse,rather than the crack‐like character associated with the aginglaw (Figure 7a). One tip of the slip zone begins to propagatefaster than the other, and eventually becomes an acceleratingslip pulse. A healing zone trails behind the slip pulse, re-strengthening the fault to a new (but lower than initial) stressstate.[44] A feature of pulse‐like nucleation is that the maxi-

mum slip speed occurs at the leading edge of the nucleationzone, in an area with little cumulative slip. Since thermalpressurization is effectively a slip‐weakening process, itsinfluence is significantly reduced for slip law friction com-pared to aging law friction (as noted by Ampuero and Rubin[2009]). Yet our numerical simulations indicate that thermalpressurization still eventually dominates weakening under arange of conditions. In Figure 7 we compare a simulation ofisothermal, drained, slip law nucleation with one that includesthermal pressurization, keeping all other parameters identical

Figure 7. Comparison of slip law nucleation (a‐c) without and (d‐f) with thermal pressurization. Weshow the half of the nucleation zone that becomes a slip pulse. Figures 7a and 7d show snapshots of slipspeed on the fault. The qualitative difference above 10−2 m/s suggests when thermal pressurization beginsto dominate fault weakening inside the nucleation zone. When vmax is greater than those speeds, thermalpressurization causes the slip pulse to narrow. The inset in Figure 7a shows the entire nucleation zone,including the side that does not develop a sustained slip pulse. Figures 7b and 7e are snapshots oflog10(v�/dc). Figures 7c and 7f are snapshots of frictional resistance to slip. Thermal pressurization sup-presses the restrengthening in the trailing region of the slip pulse. Figure 7g shows slip speed in the mid-dle of the nucleation zone over time with and without thermal pressurization. Thermal pressurization onlyslightly advances the time of the instability. Dots correlate to the snapshots shown in Figures 7a–7f.


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to the aging law simulations described above (and with a =0.6). Much like the aging law simulations, the first severalsnapshots are identical between the two cases, even as thenucleation pulse develops. Snapshots of slip speed (plots aand d) show that nucleation continues to be pulse like in bothcases, but that the pulse shape is qualitatively different athigher slip speeds (in this example, the deviation occurs forsnapshots with vmax > 10

−2 m/s, which corresponds roughlyto vcrit). With thermal pressurization, slip speed decays over amuch shorter distance behind the pulse tip. In other words,thermal pressurization leads to a smaller pulse width forcorresponding maximum slip speeds. As in aging lawnucleation, thermal pressurization causes the frictionalinstability to occur at a slightly earlier tinst, though the timeadvance is proportionally smaller at about 5 × 10–7 of thetime it takes for slip to nucleate from below steady stateconditions.[45] Figure 8 shows cumulative slip in relation to the

location of vmax and the peak stress, which defines the pulsetip. Early in nucleation, the maximum pore pressure occursat the point of maximum slip, but as the pulse accelerates,the pressure maximum quickly moves to the location ofvmax.[46] The amount of heating is a convolution of frictional

work with a diffusion kernel [e.g., Carslaw and Jaeger,1959],

T y¼0; tð Þ ¼ 12�cv

ffiffiffiffiffiffiffiffi

cth

pZ t0

� t′ð Þv t′ð Þffiffiffiffiffiffiffiffiffiffit � t′

p dt′ : ð40Þ

When vmax is stationary and continuously increasing, theintegral in equation (40) is maximal at the location ofmaximum slip. In a pulse‐like regime, however, v becomesconcentrated at the pulse tip, where little prior slip hasoccurred. In this case, the integral in equation (40) isdominated by the recent slip rate history, such that themaximum temperature is located near the current vmax(Figure 8b). Given sufficiently high slip speeds, thermalpressurization still ultimately dominates fault weakening. Atvmax > vcrit, the dominant weakening occurs behind the pulsetip, which leads to a slowing of the pulse that we consider indetail in section 4.2.[47] Because slip law nucleation is a moving pulse, we

use a slightly different definition of vcrit than in section 3.2.Instead of integrating the weakening effects over all regionsthat have weakened at prior times, here we only consider theregions weakening at the current time: and only in the activepulse. That is, we compare the magnitudes of Mtp and Mfricaccording to equations (36) and (37), but where L(t) is nowdefined as the region of x in the active pulse with negative∂tfric/∂t at time t. Figure 9a shows the determination of vcritwith this method. The jog at v ≈ 3 × 10−7 m/s corresponds tothe formation of a pulse at the end of the initial slip‐weakening phase.[48] Because of the reduced effect of thermal pressuriza-

tion relative to aging law nucleation, vcrit is higher in sliplaw nucleation. We find that, for the parameters used here,vcritslip law ≈ 100vcritaging law. While significantly higher, weemphasize that vcrit is still at or below seismic slip speeds fora range of realistic hydraulic diffusivities (discussed insection 5.5), including that which is used in the currentsimulation (chyd = 10

−6 m2/s).

Figure 8. (a) Snapshots of cumulative slip for isothermal,drained slip law nucleation. Snapshots here correspond tohalf decade increments in vmax, so every other snapshot cor-responds with those shown in Figure 7. White dots indicatethe location of vmax and black dots indicate the pulse tiplocation as defined by the stress maximum. (b) Cumulativeslip with thermal pressurization. White diamonds indicatethe location of maximum pore pressure. For clarity, markersare omitted for the half decade snapshots. (c) Temperatureand pore pressure rise on the fault. The half decade snap-shots are omitted.


13 of 25

[49] An expected consequence of rate‐ and state‐dependentfriction under isothermal, drained conditions is that slipacceleration _v is proportional to v2. In aging law nucleationwith the no‐healing approximation ( _� ≈ −v�/dc) and fixedlength nucleation ( _� ≈ −kv with constant k), this leads to anODE of the form _v = Cv2 [Dieterich, 1992]. With the sliplaw, however, behavior is more complex. Ampuero andRubin [2008] observed in their simulations of isothermalslip law pulses that slip acceleration _vmax is proportional tovmax2 and the frictional state ahead of the pulse, and satisfies

dvmaxdt

¼ Cv2max

dclnvmax�bg

dc; ð41Þ

where �bg is the frictional state ahead of the pulse and Cis an empirical constant. When thermal pressurization isincluded, _vmax deviates from equation (41), which is shownin Figure 9b. In fact, _vmax at vcrit is about double the valuepredicted by equation (41). For comparison, _vmax for anisothermal simulation with identical initial conditions isplotted; it corresponds well with equation (41). The height-ened slip acceleration at vmax > vcrit is simply a consequenceof the fact that thermal pressurization is weakening the faultfaster than in the drained case.[50] Figures 7c and 7f illustrate how thermal pressuriza-

tion affects the stress behind the pulse tip. In the isothermal,drained case, there is a stress minimum behind the tip with atotal stress drop proportional to ln vmax. Thermal pressuri-zation causes smaller stress drops for a given slip speed, orput another way, slip rates are higher for a given stresschange. This behavior is consistent with the enhancedacceleration noted in Figure 9b.

4.2. Nucleation Pulse Propagation

[51] In Figures 7 and 8, it is clear that thermal pressuri-zation affects the slip pulse propagation during slip lawnucleation. Since the distribution of shear stress and fric-tional state following nucleation influences subsequentseismic rupture, pulse propagation bears further examina-tion. Figure 8 shows that, with thermal pressurization, thepulse propagates a shorter distance than in the isothermalcase for the same values of vmax. In Figure 10a, we plotpulse position as a function of time remaining to instability,tinst; thermal pressurization clearly reduces the distance thepulse propagates. At early times (larger tinst − t), the pulsesin both cases have similar trajectories, which indicates thatthe total duration of pulse‐like nucleation is roughly equalbetween the isothermal, drained case and the case withthermal pressurization. Careful examination (not shown inFigure 10) reveals that thermal pressurization shortens thepulse duration by a tiny amount by both delaying its for-mation (by up to 10−5 of the pulse duration) and advancingthe instability (Figure 7g; about 10−4 of the pulse duration).[52] Since thermal pressurization decreases the propaga-

tion distance while keeping the pulse duration roughlyconstant, it also diminishes the pulse propagation velocity.In Figure 10b, a substantial slowing of the pulse propagationspeed is visible for vmax > vcrit. Ampuero and Rubin [2008]noted that the pulse tip propagation speed follows

vmax ¼ � dxpulsedtd�

dx

��x vmaxð Þ

; ð42Þ

where d is the cumulative slip and its gradient is evaluated atthe location of maximum slip speed vmax. This relationshipis generally valid for a translating pulse as long as the slipgradient at the tip changes more slowly than the pulsepropagates, and is shown schematically in Figure 10c. Rubinand Ampuero [2009] observe that the slip gradient changesvery little compared to _xpulse, and we find that thermalpressurization does not change that fact (Figure 10d) despitehaving a small effect on the slip gradient.[53] The reason thermal pressurization slows the pulse

propagation velocity can now be understood. Thermal

Figure 9. (a) Weakening rates for slip law nucleation, asgiven by equation (37) and presented in the same fashionas Figure 5. The critical velocity vcrit is marked. (b) Slipacceleration. As vmax approaches vcrit, slip accelerates fasterwith thermal pressurization than in the isothermal, drainedcase. The solid line shows the prediction of equation (41)[Ampuero and Rubin, 2008] given the values of vmax and�bg in the simulation with thermal pressurization. It agreeswith the isothermal result, but not the result with thermalpressurization.


14 of 25

pressurization increases slip acceleration _vmax for a givenvmax (Figure 9b). At any given time to instability (tinst − t),a greater _vmax implies that vmax must be less; if the fault isaccelerating faster yet reaches instability at nearly the sametime, the instantaneous velocity must be less. Hence, fromequation (42), a lower value of vmax implies that _xpulse issmaller with thermal pressurization.

5. Effects of System Parameters

5.1. Analytic Prediction

[54] To guide our exploration of the dependence of ther-mal pressurization on the various system parameters, wefirst write an analytical expression for thermal pressuriza-tion. Following Segall and Rice [2006], we start withequation (40), the expression for shear heating on theboundary of a diffusive body. We next assume that, at least

initially, the stress does not vary significantly (that is, thatthe stress change is much less than the total shear stress on thefault), so that for the purpose of this estimate, t ≈ m0(s − p0).Using equation (16), we obtain the pore pressure on thefault as,

p y¼0; tð Þ � L�0 �� p0ð Þ2�cv

ffiffiffiffifficth

p þ ffiffiffiffiffiffiffiffichydp �24

35 1ffiffiffi

pZ t0

v t′ð Þffiffiffiffiffiffiffiffiffiffit � t′

p dt′ ; ð43Þ

in which the bracketed term serves as a coefficient thatcontrols the intensity of pressurization resulting from anarbitrary slip rate history. As mentioned in section 3.2, thecritical velocity vcrit is a convenient measure of the inten-sity of thermal pressurization: if thermal pressurization isstrong, it dominates at lower slip speeds. The definition ofvcrit in equation (37) includes nucleation zone geometry,

Figure 10. (a) Slip law pulse position at times before tinst. Dots correspond to snapshots in Figure 7. Theearly backwards motion is the stress maximum moving inward during the initial slip‐weakening phase.The gray dot corresponds to vcrit. (b) Pulse propagation velocity. The roll‐off when tinst is imminentindicates a loss of numerical precision. (c) Schematic illustrating equation (42), the relationship betweenpulse propagation velocity, slip speed, and slip gradient. (d) Thermal pressurization satisfies equation (42),which is shown by the solid line.


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which complicates analysis of equation (43). A simplerpointwise definition that we adopt in this section is

vcrit ¼ vmax when �0 _p ¼ � �� p0ð Þ _� at the location of vmax:ð44Þ

All references to velocity in this section refer to vmax; forsimplicity, we shall omit the subscript max.

5.2. Dependence on a and b: Aging Law

[55] With the aging form of the state evolution law (5),analytical expressions [Dieterich, 1992; Rubin and Ampuero,2005; Ampuero and Rubin, 2008] can be written to describethe slip history of an isothermal nucleation zone, which maythen be used to approximate v(t) in equation (43). Segall andRice [2006] present that analysis for a nucleation zone offixed length, which they took to be Lmin (equation (25)). Todefine the critical slip speed vcrit, they compare the rate ofthermal weakening −m0 _p to the elastic unloading rate _� =−d(kd)/dt that would occur in the absence of thermalpressurization.[56] We derive a similar estimate, but informed by the

results of Ampuero and Rubin [2008] for aging law nucle-ation. The maximum slip speed in aging law nucleation isgiven by

v tð Þ ¼ 1vi� H a; b; dcð Þt

a

� ��1; ð45Þ

where vi is the initial slip speed and H is a quantity thatcontrols the slip acceleration (larger H leads to earlier insta-bility and greater acceleration prior to instability). Based onthe results of Rubin and Ampuero [2005], an estimate of H is

H ¼

� 2ð Þb

dc

fora

b� � 2

2dcb� að Þ a

bfor

� 2

� ab< 1 :

8>><>>: ð46Þ

The expression forH for low a/b is slightly different than thatof Rubin and Ampuero [2005]; using numerical methods,they found H ≈ 0.3781b/dc, which differs from equation (46)by 4%. We chose to use algebraic expressions in p formathematical simplicity, since these expressions are only

approximations. For the special case of a/b = (p − 2)/p,equation (46) simplifies to H = a/dc, which is Dieterich’s[1992] expression for nucleation of a zone of fixed lengthLmin under the no‐healing approximation. Segall and Rice[2006] use this value of H in their estimate of vcrit. Substi-tuting equation (45) into equation (43) and evaluating theintegral leads to

p y¼0; tð Þ � L�0 �� p0ð Þffiffiffiffiffiav

p

�cvffiffiffiffifficth

p þ ffiffiffiffiffiffiffiffichydp � ffiffiffiffiffiffiffiHp arctanffiffiffiffiffiffiffiffiffiffiffiffiv

vi� 1

r: ð47Þ

In most cases, v � vi so the arctangent evaluates to p/2. Wenote from equation (45) that

dv

dt¼ H

av2 ; ð48Þ

so the pressurization rate is therefore

dp

dt

��y¼0;tð Þ

� L�0 �� p0ð Þffiffiffiffiffiffiffiffiffiffiffi

Hv3

p

4�cvffiffiffiffifficth

p þ ffiffiffiffiffiffiffiffichydp � ffiffiffiap ; ð49Þ

which is exactly equation (55) of Segall and Rice [2006]. InFigure 11, we demonstrate that equation (49) is a very goodestimate by comparing it with _p from the simulation shown inFigure 6 (with a/b = 0.8).[57] To estimate vcrit, we compare equation (49) to the

frictional weakening rate _�(s − p0). This approach neglectsthe feedback of pressurization into slip acceleration, butremains instructive. In the absence of inertial effects, fric-tional weakening is equal to the decrease in driving stress.Under drained conditions

�� p0ð Þ d�dt ¼ �d

dtk�ð Þ ¼ �� dk

dt� kv : ð50Þ

In general, the stiffness of a slipping zone is inverselyproportional to its length; that is, k / G/2L, where L is thehalf length of the zone. Assuming unit proportionality,equation (50) becomes

�� p0ð Þ d�dt ¼G

2L

�

L

dL

dt� v

� �: ð51Þ

The critical half length for rate/state friction is

L ¼ Gdc2�eff f a; bð Þ ; ð52Þ

where f (a, b)is different for Lmin, Ln, or L∞. We define vcritas the slip speed, v, for which m0 _p = −(s − p0) _�. Thenucleation zone’s growth rate _L is proportional to the slipspeed inside the zone (∼v) and slip gradient dd/dx (similar toequation (42)); since ∣dd/dx∣ � d/L near the tip of a slipzone, we find d _L/L � v. With that approximation, equations(49), (51), and (52) yield

vcrit � a f a; bð Þ½ �2

H


p þ ffiffiffiffiffiffiffiffichydp �Ldc�20

24

352

ð53Þ

for which we can select different values of f (a, b) and H forthe different aging law nucleation regimes.

Figure 11. Comparison of dp/dt predicted by equation (49)(gray) and the numerical results for an aging law simulationwith a/b = 0.8 (black). The dot corresponds to vcrit.


16 of 25

[58] Segall and Rice [2006] chose H = a/dc and f(a, b) =b − a (corresponding to Lmin) to obtain

vSR06crit �1

dc

4�cv b� að Þ ffiffiffiffifficthp þ ffiffiffiffiffiffiffiffichydp �L�20

24

352

: ð54Þ

We note, however, that the value ofH given by equation (46)is more appropriate in the later phase of nucleation. We alsochoose f(a, b) as appropriate for the phase of nucleation andthe a/b regime. The initial slip phase has half length Lmin,but slip then localizes approaching dimension Ln. For a/b >0.5, the nucleation zone then grows toward half length L∞.This evolution can be seen in Figure 12, which correspondsto the isothermal, drained simulation shown in Figure 2.Following equation (46) and Rubin and Ampuero [2005], weuse the following expressions for f (a, b):

f a; bð Þ ¼

b

8for

a

b� 1

2; for L ¼ L�

b� að Þ22b

fora

b 1

2; for L ¼ L∞ :

8>><>>: ð55Þ

As with the approximations for H (see equation (46)), wesubstitute an approximate expression that is algebraic in p andpiecewise continuous over ab−1 (compare to equation (34) forLn). The expressions in equation (55) correspond to nucle-ation zones that have attained their final sizes, which is notnecessarily the case when vmax = vcrit but still provides a

reasonable estimate. Combining equations (46), (53), and(55) yields

vagingcrit ¼1

dc


p þ ffiffiffiffiffiffiffiffichydp �L�20

24

352

3ab

64 � 2ð Þ for 0 <a

b� � 2

b3

32 b� að Þ for

� 2

� ab� 1

2

b� að Þ32b

for1

2� a

b< 1 :

8>>>>>>><>>>>>>>:

ð56Þ

Figure 13 compares numerical values of vcrit to predictionsfrom (56), and shows that they are in general agreement. Theanalytical prediction is high by a factor of ∼2, which weattribute to the fact that it neglects the feedback of increases inp on frictional strength.

5.3. Dependence on a and b: Slip Law

[59] Figure 14 presents numerical values of vcrit for vari-ous values of a and b with the slip law, using the nominalvalues for all other parameters. Clearly, thermal pressuri-zation exhibits in two markedly different regimes. For a/b ⪅0.65, thermal pressurization dominates at slow slip speedsand exhibits strong dependence on a and b. In this regime, ahealing front does not develop behind the tip of the slipzone, so it cannot be described as “pulse like.” For a/b ^0.65, the nucleation zone develops a healing front and isthus pulse like. In this regime, the critical velocity vcrit ishigh and only weakly depends on a and b over a wide rangeof a/b. Qualitatively, the two regimes are a consequence ofthe competition between the speed of pulse propagation andthe rate of thermal pressurization. To first order, the slip

Figure 12. Weakening rate during isothermal nucleation,which is described by equation (51). Straight lines corre-spond to the decrease in stress due to slip in fixed widthnucleation zones. The numerical result of the isothermal,drained nucleation zone of Figure 2 is shown. The nucle-ation zone initially weakens in a zone of length 2Lmin, thencontracts toward a length of 2Ln, and finally expands towarda length of 2L∞.

Figure 13. Critical velocity vcrit for various a and b in aginglaw nucleation. Black dots are numerical results. The grayline is the prediction of equation (54) [Segall and Rice,2006]. Gray dots are the predictions of equation (56). Mul-tiple points at a given b − a reflect different values of a/bfrom 0.263 to 0.95.


17 of 25

pulse propagation speed is proportional to (1 − a/b)−2 [Rubinand Ampuero, 2009] (though its second‐order dependenceon a and b is considerably more complicated). Hence lowervalues of a/b lead to slower pulses, which do not propagatefast enough to “outrun” thermal pressurization. On the otherhand, high values of a/b lead to rapidly propagating slippulses for which the zone of greatest weakening occurs in anarea with little prior shear heating.[60] We note that the pointwise definition for vcrit (44) and

the whole pulse definition (37) can differ significantly forfast pulses. In this case, the point of maximum pressurizationfollows significantly behind the location of vmax (Figure 8b).Consequently, when equation (44) is satisfied, most of thepulse behind x(vmax) is already dominated by thermal pres-surization. We have found that the pointwise estimate of vcritcan be as much as 10 times higher than the whole pulsedefinition for large a/b. For a/b = 0.8, the pointwise value ofvcrit is about 5 times the whole pulse value.[61] Because of the highly nonlinear trajectories of v and �

with slip law friction, we have been unable to developanalytical predictions of vcrit. For the quasi‐stationarynucleation zones of the low‐a/b regime, we cannot followthe same method developed in equations (40)–(56) abovesince explicit expressions for v(t) and L are not known. Forthe pulse‐like nucleation, an appealing approach is to treatthe pulse arrival as a Heaviside function, v(t) = vH(t), toderive an expression for p(y = 0, t). Numerical results,however, indicate that the velocity history is considerablymore complex than a Heaviside function, with a smallamount of precursory acceleration followed by finiteacceleration during the risetime.[62] Despite the lack of an analytical prediction for ther-

mal pressurization with slip law friction, we have found thatthere is a systematic dependence of vcrit on parameters aand b. Using a least squares fit to the numerical estimates of

vcrit in terms of a, b, and (b − a), we find for the low‐a/bregime that

vlowcrit �2 b� að Þ3

adc



24

352

ð57Þ

and, for the high‐a/b regime,

vhighcrit �200a9 b� að Þ

b8dc



24

352

: ð58Þ

These empirical predictions are shown in Figure 14.

5.4. Slip Weakening Distance dc[63] The slip weakening distance dc is the amount of slip

necessary for frictional state to evolve following a change invelocity. During nucleation, this controls the slip distanceover which a fault transitions from a high‐strength, low‐speed condition to a low‐strength, high‐speed condition. Italso influences shear heating by controlling how much slipmust occur for a given decrease in frictional resistance.Higher values of dc cause the sliding to occur at high frictionfor longer slip such that more frictional work is done. Fur-ther, large values of dc correspond to a modest frictionalweakening rate, thereby allowing thermal pressurization todominate earlier. In equation (56), vcrit / dc−1, which agreeswith simulations for small dc shown in Figure 15.[64] Interestingly, however, the dependence of vcrit chan-

ges as dc gets larger. We attribute this behavior to the factthat large values of dc cause thermal pressurization todominate during the initial slip phase rather than during thegrowing crack phase. In the general aging law prediction forvcrit, equation (53), f(a, b) = b − a and H = a/dc during the

Figure 14. Critical velocity vcrit for various a and b in sliplaw nucleation with thermal pressurization. Black dots arenumerical results. Gray triangles and dots are empirical pre-dictions given by equations (57) and (58), respectively. Withthermal pressurization, low a/b nucleation is not pulse like.

Figure 15. Critical velocity vcrit for various dc in aging lawnucleation with thermal pressurization. Black dots arenumerical results. The upper gray line is the prediction ofequation (54) [Segall and Rice, 2006] while the lowergray line is the prediction of equation (56).


18 of 25

initial slip‐weakening phase. These expressions for f and Hare precisely the assumptions made by Segall and Rice[2006] and lead to equation (54), which is plotted inFigure 15. As aging law nucleation transitions from itsinitial slip‐weakening phase (width 2Lmin) to its growingcrack phase (width approaching 2L∞), the rate/state fric-tional weakening rate’s dependence on vmax exhibits a kinkas the nucleation zone length changes rapidly (Figure 12).Simulations with dc values that cause vcrit to occur duringthe transition between nucleation zone lengths thus reflectthe kink in (s − p) _� in Figure 12. If laboratory values of dc(10–100 mm) are applicable at seismogenic depths, how-ever, the large values of dc shown in Figure 15 would notrepresent the behavior of natural faults.

5.5. Hydraulic Diffusivity

[65] Of all of the material properties relevant to thermalpressurization, shear zone hydraulic diffusivity chyd is likelythe least well known. For that reason, we tested a broadrange of values for both the aging and slip laws. We expectthat higher chyd will reduce the effect of thermal pressuri-zation, since diffusion reduces excess pore pressure in thefault zone. We further expect there to be a limiting value ofvcrit; even as chyd → 0, thermal diffusion limits the rate ofpressurization.[66] Values of both pointwise and whole zone/pulse vcrit

for several numerical simulations are plotted in Figure 16a,and they capture the anticipated dependence on hydraulicdiffusivity. For aging law nucleation, the numerical point-wise vcrit values agree well with the analytic estimate givenby equation (56). Estimates with the whole nucleation zonelead to larger vcrit because they include regions that haveweakened by rate/state friction but have not undergone asmuch shear heating. For slip law nucleation, pointwise vcritis roughly three orders of magnitude larger than with theaging law, which is an expected consequence of pulse‐likebehavior. Unlike aging law nucleation, pointwise values ofvcrit are larger than whole zone values because the formerexcludes the portion of the fault with the greatest shearheating (see section 5.3).[67] As mentioned in sections 3.1 and 4.1, thermal pres-

surization advances the time of the instability tinst relative tothe isothermal case by a modest amount (for a discussion oftinst in the drained isothermal limit, see Ampuero and Rubin[2008]). Increasing hydraulic diffusivity reduces the effectof thermal pressurization, and therefore reduces the timeadvance of instability (Figure 16b). For both aging and slipstate evolution laws, there is a maximum advance in tinstin the undrained limit. While a few hundreds of secondslong, both are relatively small compared to the total nucle-ation time: about 2 parts per million for aging law nucleationand half that for the slip law. Hence thermal pressurizationwill not significantly bias processes that depend on nucle-ation time, even for faults with extremely low hydraulicdiffusivity.

5.6. Other Parameters: m0, L, r, cv, and (s − p0)[68] Although values of the remaining material properties

are fairly well constrained from laboratory observations, wetested their influence on thermal pressurization. Equation (43)indicates that the pore pressure increase is directly propor-tional to the nominal friction coefficient m0, and with theaging evolution law, that leads to a prediction of vcrit / m 0−4in equation (56). Qualitatively, lower values of m0 corre-spond to less frictional work for the same amount of slip,thereby reducing thermal pressurization. We find (Figure 17a)that the dependence of vcrit on m0

−4 is borne out in numericalsimulations. Thermoelastic and poroelastic properties exert astrong influence on thermal pressurization (section 5.5).Equation (56) predicts that vcrit scales with (rcv/L)

2.Although this nondimensional group is relatively well con-strained [Rice, 2006] near the value of 0.28 used throughoutthis paper, we verify its influence on vcrit in Figure 17b.[69] As indicated by dimensional analysis (32) and, at

least for the aging law, the analytical prediction for vcrit (56),there is no dependence of vcrit on ambient effective stress

Figure 16. (a) The effect of hydraulic diffusivity chyd onvcrit. All other parameters have nominal values (Table 1).The thermal diffusivity is cth = 10

−6 m2/s, so D ranges from10−4 to 106. Results from aging law simulations are com-pared to equation (56). (b) Relative advance of the frictionalinstability time over the isothermal instability time. Thequantity shown is (tinst − tinstisothermal)/tinstisothermal. Increasingthe strength of thermal pressurization advances the time ofthe frictional instability very modestly. Dots indicate aginglaw nucleation; boxes indicate slip law nucleation.


19 of 25

(s − p0), which is confirmed by numerical results. Oneexception exists: nucleation is affected at extremely lowambient effective normal stress. Since shear heating ceaseswhen m(s − p0 − p) = 0, the ambient effective stress imposesan upper bound on the stress drop associated with fault slip.We have not yet investigated models in which a total stressdrop occurs during the nucleation phase. Recent studies [e.g.,Audet et al., 2009] suggest that near‐lithostatic pore pressureis possible at depth in subduction zones, so the issue warrantsfurther consideration.

6. Discussion

6.1. Role of Thermal Pressurization During Nucleation

[70] Our analysis suggest shear heating‐induced thermalpressurization is significant (and under many circumstancesmay be dominant) during the late stages of earthquakenucleation. During the early phase of nucleation on a steadystate velocity weakening fault, however, its weakening rate isinsignificant compared to the state evolution effect in friction.

For that reason, rate/state friction remains necessary to initiateslip instabilities on uniformly loaded faults. In effect, thermalpressurization requires that slip accelerates to a sufficientlyhigh speed that the rate of heat production overwhelms dif-fusion. Hence thermal pressurization is unlikely to lead to aslip instability on a uniformly loaded, slowly slipping, steadystate velocity strengthening fault. Segall and Rice [2006]addressed that issue with a linear stability analysis; theyshowed that even for very small chyd, perturbations fromsteady state v are stable for a > b. Regardless of the frictionalregime, our analysis indicates that thermal weakening effectsshould bemodeled during fault slip at moderate to high quasi‐static slip speeds. If, as we suggest, thermal pressurization issignificant before dynamic instability, we expect it to beactive during both large and small earthquakes. That infer-ence is consistent with the analysis of Ide and Beroza [2001],which shows little dependence of apparent stress drop onmagnitude for −3 < Mw < 8.

6.2. Dilatancy as a Possible Mitigating Effect

[71] While our results argue for the significance of ther-mal pressurization, we have not considered other mechani-cal processes that may mitigate its effect. One effect notexamined in this paper is dilatancy of shear zone materials,which can counteract the pore fluid expansion and diminishthermal pressurization [Segall et al., 2010]. The interactioncan, however, be complex; dilatancy increases the effectivenormal stress, which in turn leads to enhanced heat pro-duction [Garagash and Rudnicki, 2003]. Moreover, dilat-ancy is likely to be dominant only at low effective normalstress. For a given dilatancy rate (that is, rate of increase inporosity) D _�, Segall et al. [2010] show that the pressureboundary condition at the fault is (letting h → 0 but keepingh _� nonzero)

@p

@y

��y¼0

¼ h_�

2eff chyd: ð59Þ

The constitutive law for dilatancy of Segall and Rice [1995](motivated by experiments of Marone et al. [1990] is

D� ¼ �� ln v0�dc

; ð60Þ

with laboratory values of � for gouge of ∼10−4. Normalizingas in section 2.3, we obtain a pore pressure boundary con-dition of

@p̂

@ŷ

��ŷ¼0

¼ �E 1�̂

d�̂

dt̂; ð61Þ

where the “dilatancy efficiency” (called Ep by Segall et al.[2010]) is given by

E � �2eff �� p0ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffih2v∞chyddc

s: ð62Þ

In equation (61) it becomes apparent that large effectivenormal stress reduces the effect of dilatancy. The ratioE=FD1/2 measures the relative importance of dilatancy andthermal pressurization. For the nominal parameters used inthis study, E=FD1/2 = 0.05, which indicates that dilatancy is

Figure 17. (a) The effect of nominal friction m0 on vcrit.(b) The effect of (L/rcv) on vcrit. In both plots, dots are numer-ical results and lines are predictions of equation (56).


20 of 25

insignificant compared to thermal pressurization at higheffective normal stress. Numerical simulations confirm thatresult; for example, a simulation with � = 10−4 but otherwiseidentical to the one shown in Figure 2 yields the same vcrit,although the induced pore pressures are ∼10% lower. Anextensive study of the interaction between dilatancy andthermal pressurization is in progress [Bradley and Segall,2010].

6.3. Distributed Shear

[72] Large shear zone thickness h suppresses thermalpressurization because distributing frictional work over alarger volume diminishes the temperature rise. As discussedin section 2.1, a simple dimensional argument indicateswhen modeling distributed shear is necessary. Laboratoryexperiments suggests that h and dc may be related; re-arranging equation (3), we find that distributed shear mustbe modeled for

v tð Þ^ 16cdc

dch

� �2: ð63Þ

Marone and Kilgore [1993] propose h ≈ 100dc based onlaboratory friction experiments using simulated fault gougebetween steel sliding blocks. Chambon et al. [2002] shearedquartz sand and observed shear bands of comminuted par-ticles that were ∼60 times larger than the observed dc. Rice[2006] notes that shear bands in granular media are typically10–20 particles wide, and that dc likely represents particlesize. For dc = 100 mm and h = 2 mm, equation (63) suggeststhat distributed shear must be considered for v > 4 × 10−4 m/s.For dc = 10 mm and h = 200 mm, we find v > 0.004 m/s. Sincethese slip speeds are comparable to vcrit for a zero‐width shearzone, the issue of distributed shear demands further study.[73] To investigate the effect of shear zone thickness, we

performed simulations of aging law nucleation with dc =100 mm and various shear zone thicknesses h by employinga newer code that integrates separate temperature and porepressure diffusion grids and is capable of distributed shearheating [Bradley and Segall, 2010]. The new code has beenvalidated against the results shown in the rest of this paper.

We assume a Gaussian distribution of shear strain rateacross a zone with RMS half width h/2, given by

d� x; y; tð Þdt

¼ffiffiffi2

pv x; tð Þffiffiffi

ph

exp � 2y2

h2

� �; ð64Þ

which accommodates a total slip speed v(t). Figure 18 showsthe effect of distributed shear. For h ≤ 10dc, distributed shearhad little effect on vcrit. For 10dc < h < 20dc (the favoredrange of Rice [2006]) increasing h leads to increases of vcritof only 10% to 20%, yet the along‐strike localization of slipshown in section 3.1 is greatly reduced. For larger shearzone thicknesses, the influence of thermal pressurizationweakens dramatically; for h ^ 50dc, thermal pressurizationstill dominates fault weakening at the location of fastest slip(so vcrit is defined), but most of the nucleation zone remainsdominated by rate/state friction. For h ^ 150dc, thermalpressurization fails to dominate weakening anywhere on thefault before radiation damping becomes dominant, althoughit is still a significant contribution to the fault’s resistance toshear.

6.4. Effect on Seismicity

[74] An important implication of our simulations is thatstatistical models of seismicity rates, like that of Dieterich[1994], are not significantly affected by thermal pressuri-zation. Such seismicity models depend on the time fault sliptakes to become unstable. The time to instability for slipnear steady state is

tinst � t ¼ 2bdc

b� að Þvmax tð Þ ð65Þ

for the aging law with 0.4 < a/b < 1 [Ampuero and Rubin,2008], and it is of comparable magnitude (but slightlysmaller and far more difficult to estimate analytically) forthe slip law [Rubin and Ampuero, 2009]. For “locked” faults(log vmax < −12), the time to instability (tinst − t) is severalorders of magnitude longer than the ∼1000 s advance thatresults from thermal pressurization (Figure 16b). For thatreason, it is not necessary to include thermal pressurizationin seismicity rate models that use rate/state friction (forexample, the regional earthquake simulator of Dieterich andRichards‐Dinger [2008]).[75] The small nucleation zones we observe in our simu-

lations may have implications for detection of the nucleationphase and for estimates of the minimum earthquake mag-nitude. Recently, researchers have attempted to record theearliest seismic phases of earthquakes by placing strain-meters and seismometers close to faults in deep mines[Boettcher et al., 2009] or in the SAFOD borehole [Zobacket al., 2010]. Under isothermal rate/state friction, nucleationzones have a minimum size at seismic slip speeds. For theaging state evolution law with 0.5 < a/b < 1, its width isnearly L∞ (which is perhaps 1.27 to 60 times Lmin). Withthermal pressurization, however, the active slipping zone ismuch smaller when it reaches seismic slip speeds, and willtherefore be much harder to observe with near‐source instru-ments. For isothermal slip law nucleation, the actively slip-ping region is already small (a fraction of Lmin) thermalpressurization does not make detection of precursory slipsignificantly more challenging. In the case of earthquakes that

Figure 18. Dependence of vcrit on shear zone thickness h foraging law nucleation with parameters otherwise identical tothose in Figure 2 (dc = 10

−4). For h ^ 5dc, vcrit increaseswith h, which shows that distributed shear

shear heating induced thermal pressurization during earthquake nucleation · 2019. 11. 12. · r t_...

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